For which Artin algebras $ A$ is the stable module (that is the module category modulo projectives) category of finitely generated modules abelian? If you like you may take rings that are not Artin algebras or look at all modules instead of just finitely generated modules.
Non-trivial examples should include $ A=K[x]/(x^2)$ or $ A$ being a hereditary algebra of Dynkin type $ A_2$ or $ A_3$ with linear oriented orientation. I do not know many examples but all had dominant dimension at least one. Is there one with dominant dimension zero? Is there a nonselfinjective example of infinite global dimension?
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